Your probability is 0.80, but remember, this is the area in the middle, not from one end of the normal distribution curve.

Hint: Your 80% is centered around 1.9, so what are the probabilities of each end that is NOT within the safety limits. After that, all you need is the invNorm function on your calculator.

EDIT: Woah, on second thought, I'm not sure what method you and the answer sheet use. Maybe my way isn't the way you were taught. However, I know for sure it works.

Yeah I dont know the method with the "invNorm function".
Regarding the hint "Your 80% is centered around 1.9", yes i have taken that into account. Since the rane as i stated was -b/0.15< Z < b/.15. I first formed an equation for the probability where Z< b/0.15. Then I formed a probability of Z< -b/0.15. Now I deduct the 2 probabilities(their equations actually) to get the actual probability, the resultant equation in b(which like you said is centered) Now this resultant equation is 0.8. So I solve for b. But apparently it is wrong.